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GIFT  OF 
ENGINEERING  LlBfiARY 


REPORT  OF  THE 
CO-INSURANCE  COMMITTEE 
TO  THE 
BOARD  OF  FIRE  UNDERWRITERS  OF  THE  PACIFIC 

ON 

PERCENTAGE  CO-INSURANCE 

AND  THE 

REI.ATIVE  RATES  CHARGEABLE  THEREFOR 

ALSO   ON   THE   COST   OP 

CONFLAGRATION  HAZARD  OF  LARGE  CITIES 


San  Francisco,  California 
September,  igos. 


REPORT  OF  THE 

CO-INSURANCE  COMMITTEE 

TO  THE 

BOARD  OF  FIRE  UNDERWRITERS  OF  THE  PACIFIC 

ON 

PERCENTAGE  CO-INSURANCE 

AND   THE 

REI.ATIVE  RATES  CHARGEABLE  THEREFOR 

ALSO   ON   THE   COST   OF 

CONFLAGRATION  HAZARD  OF  LARGE  CITIES 


San  Francisco,  California 
September,  1905. 


S3B4 


GIFT  or 

ENGINEERING  LIBRARY 


3JC. 


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To  ^Tie  Executive  Committee,  Board  of  Fire  Underwriters 

of  the  Pacific : 
Gentlemen  : 

Your  sub-Committee  on  the  subject  of  Co-insurance 
being  adopted  for  San  Francisco  beg  to  report : 

After  careful  consideration  of  what  has  been  written 
on  the  subject  and  all  the  data  obtainable,  they  are  of  the 
opinion  that  up  to  the  present  time  all  the  different  rate 
reductions  for  co-insurance  are  arbitrary  to  a  greater  or 
less  extent,  and  that  until  other  statistics  are  utilized 
than  heretofore,  they  will  so  continue. 

As  to  cities  and  towns  where  sufficiently  reliable  data 
are  obtainable  as  to  losses  and  values,  there  is  no  reason 
why  the  value  of  the  percentage  co-insurance  clause 
should  not  be  ascertained  within  a  small  degree  of  exact- 
ness. Those  who  have  given  or  who  will  give  careful  con- 
sideration to  the  question  will  find  that  nothing  approach- 
ing facts  regarding  value  of  co-insurance  can  be  learned 
without  loss  to  value  of  property — that  is  the  foundation 
stone;  it  has  been  claimed  by  some  that  loss  to  insurance 
is  good  enough,  especially  as  any  error  on  such  a  basis 
of  calculation  would  result  in  favor  of  the  insurance 
companies,  but  it  is  not  wise  to  let  non-co-operating  com- 
panies have  so  wide  a  margin  in  their  favor,  and  as  the 
amount  of  insurance  carried  is  a  movable  quantity  at  the 
will  of  the  assured,  it  will  readily  be  seen  that  facts  can- 
not be  learned  therefrom;  advocates  of  this  plan  argue 
that  values  of  property  as  learned  from  proofs  of  loss 
are  not  altogether  reliable;  that  as  to  individual  losses 
is  occasionally  likely  to  be  the  case,  but  in  a  large  number 
of  losses  under-  and  over-statements  of  values  will  adjust 
themselves  very  closely,  and  after  everything  that  may 
be  said  against  the  reliability  of  such  values  of  prop- 
erty, no  better  method  of  obtaining  such  values  has  been 
suggested. 


Your  Committee  are  ambitious  enough  to  wish  they 
could  settle  the  value  of  co-insurance  over  the  entire 
United  States,  but  their  opportunities  of  getting  sta- 
tistical information  are  too  limited,  having  regard  to 
the  area  under  their  jurisdiction,  but  they  are  not  with- 
out strong  hope  that  this  report  may  pave  the  way  for 
Eastern  organizations  to  co-operate  on  similar  lines  and 
compile  the  data  gathered  so  as  to  make  an  average  that 
could  be  applied  over  an  important  part  of  the  United 
States.  One  table  would  not  answer  because  in  the  opin- 
ion of  your  Committee  the  value  of  co-insurance  fluc- 
tuates according  to  construction,  climatic  conditions, 
water  supply,  fire  department,  etc.  It  would,  therefore, 
be  necessary  to  divide  cities  and  towns  into  three  or 
more  groups  and  have  loss  to  value  and  other  data 
secured  as  to  a  sufficient  number  of  each  group  to  form 
a  fair  average  of  the  whole  of  such  group.  All  towns  of 
inferior  construction,  those  with  inadequate  water 
supply  or  fire  department,  should  be  thrown  out,  co-insur- 
ance being  of  little  value  in  such  towns,  and  it  is  of  no 
value  in  the  isolated  risk  unless  under  superior  fire 
protection  or  sprinkler  equipment. 

The  question  of  mandatory  co-insurance,  eighty  per 
cent  co-insurance  only,  or  leaving  the  percentage  open 
and  fixing  rate  accordingly  was  considered  and  having 
regard  to  the  objection  of  some  people  to  compulsory 
percentage  of  co-insurance  and  the  enactment  of  anti- 
co-insurance  laws  in  some  States,  it  was  deemed  expedi- 
ent to  recommend  the  open  percentage  method  in  order 
that  the  assured  can  select  for  himself  what  he  wishes 
to  purchase  and  companies  can  charge  accordingly. 

The  only  available  statistics  of  loss  to  value  obtain- 
able by  this  Committee  are  those  relating  to  San  Fran- 
cisco fires  contained  in  proofs  of  loss  in  various  offices. 
Of  course,  similar  figures  could  be  obtained  as  to  a 
number  of  other  cities  on  the  Coast,  but  in  the  opinion 


of  the  Committee  such  other  cities  would  belong  to  a 
different  class  to  that  in  which  San  Francisco  should  be 
placed.  It  was  decided  to  employ  Mr.  Albert  W.  Whit- 
ney, Professor  of  Mathematics  of  the  University  of 
California,  and  almost  without  exception  all  the  leading 
companies  gave  him  the  opportunity  of  examining  their 
San  Francisco  proofs  of  loss  for  the  five  years  1899  to 
1903,  from  which  the  record  of  5642  fires  was  tabulated. 
From  these  facts  under  the  instructions  of  the  Committee, 
Mr.  Whitney  made  certain  classifications  and  calcula- 
tions as  shown  in  his  report  attached  hereto. 

Brick  special  hazards  were  not  taken  into  account,  as 
the  number  of  important  fires  is  too  limited  to  be  of 
material  value.  Brick  dwellings  are  not  separately 
dealt  with  as  there  are  not  enough  of  them  in  San  Fran- 
cisco to  form  a  class  by  themselves. 

We  are  aware  that  the  Universal  Mercantile  Schedule 
adopts  fifty  per  cent  co-insurance  as  the  basis  from 
which  to  increase  or  reduce  rates,  but  as  there  does  not 
appear  to  be  any  evidence  that  tabulation  of  facts 
demonstrated  that  fifty  per  cent  was  the  average  insur- 
ance carried  on  which  companies  have  been  making  their 
experience  without  co-insurance,  the  Committee  can  see 
no  reason  for  recommending  any  figure  as  a  base  line 
for  percentage  co-insurance  rates,  other  than  the  actual 
experience  as  to  average  insurance  to  value  heretofore 
carried,  and  we  are  not  experimenting  in  doing  so  be- 
cause the  same  results  as  to  profit  and  loss  heretofore 
obtained,  or  a  little  better,  can  be  looked  for  in  the 
future,  provided  we  adhere  to  past  experience  as  to  per 
cent  of  insurance  carried. 

The  relative  or  percentage  co-insurance  rate  for  San 
Francisco  based  on  the  five  years'  experience  shown  in 
Table  17  of  Mr.  Whitney's  report  should  produce  the 
same  underwriting  results  as  in  past  years,  with  a  prob- 
able improvement  in  years  of  general  business  depression 


(for  reasons  hereafter  stated)  by  using   present    rates 
as  a  basis  to  calculate  the  percentage  co-insurance  rates. 

An  important  point  to  be  considered  is  that  the  five 
years'  losses  in  San  Francisco  on  which  results  are  fig- 
ured do  not  contain  any  conflagration  losses,  so  that  the 
question  of  a  loading  being  necessary  should  be  con- 
sidered. As  stated  heretofore,  our  opinion  is  that  the 
value  of  co-insurance  has  to  be  grouped  as  to  three  or 
more  classes  of  cities  and  towns ;  therefore,  conflagration 
hazards  must  be  similarly  dealt  with,  and  taking  (for 
the  purpose  of  illustrating)  as  Class  1,  cities  showing 
over  200,000  population,  as  per  1900  census,  we  find  nine- 
teen such  cities,  and  the  census  returns  of  those  cities 
for  the  last  six  decades  are  1,591,588  for  1850,  2,926,732 
for  1860,  4,159,425  for  1870,  5,586,268  for  1880,  7,858,595 
for  1890,  and  11,795,809  for  1900.  This  would  make  a 
mean  average  annual  population  of  5,444,943.  The 
premium  income  (for  1900)  of  eleven  of  these  cities 
(having  a  population  in  1900  of  8,086,649)  sufficiently 
distinctive  and  remote  from  one  another  to  form  a  good 
average  of  the  whole,  shows  the  average  annual  premium 
per  capita  to  be  $3.77.  Of  course,  the  average  rate  has 
varied  during  the  different  periods,  but  not  sufficiently 
to  materially  affect  final  results. 

An  arbitrary  period  had  to  be  taken  for  figuring  con- 
flagration losses  and  it  was  decided  to  take  a  period  of 
fifty  years.  In  these  nineteen  cities  during  the  past  fifty 
years  we  have  had  serious  conflagrations  in  Chicago, 
Boston  and  Baltimore,  causing  an  insurance  loss  of 
$180,116,620.  The  fifty  years'  premium  income  based 
on  $3.77,  the  1900  per  capita,  being  $1,026,371,755,  and 
the  fifty  years'  conflagration  loss  to  insurance  com- 
panies on  such  premium  income  being  $180,116,620,  we 
find  17.55  per  cent  of  premium  is  the  cost  of  conflagration 
losses  in  cities  at  the  present  time  having  a  population 
of  over  200,000,  and  dividing  the  average  annual  con- 


flagration  loss  by  the  average  annual  population,  we 
find  the  conflagration  loss  to  be  $0.66  (sixty-six  cents) 
annually  per  capita. 

The  Committee  realize  that  the  question  of  conflagra- 
tion hazard  was  not  referred  to  them,  but  it  is  so  closely 
allied  to  the  co-insurance  question  that  attention  had  to 
be  called  to  it  in  this  report. 

Before  deciding,  however,  that  any  loading  is  needed 
to  present  rates  for  conflagration  hazard,  it  would  be 
necessary  to  ascertain  the  insurance  losses  in  the  nine- 
teen cities  referred  to.  The  losses,  less  the  conflagra- 
tions named,  for  the  past  five  years  or  any  other  period, 
should  be  secured  and  the  average  annual  per  capita  loss 
ascertained,  add  the  annual  conflagration  loss  of  $0.66 
per  capita,  then  compare  the  result  with  the  average 
annual  per  capita  premium  paid  for  the  corresponding 
years,  and  it  would  then  be  shown  whether  rates  needed 
grading  up  or  down,  or  whether  present  rates  are  fair 
to  all  interests  involved.  This  method  of  calculation 
could  be  applied  to  any  one  city  or  group  of  cities. 

Another  point  to  be  considered  is  that  with  reduced 
rates  on  account  of  co-insurance,  the  public  would  be  apt, 
especially  in  the  better  class  of  risks,  to  carry  a  larger 
amount  of  insurance,  and  that  under  such  conditions  a 
larger  percentage  of  insurance  to  value  would  be  in- 
volved in  conflagration  losses. 

The  figures  as  to  conflagration  losses  and  premiums 
in  large  cities  were  obtained  through  the  courtesy  of 
General  Agent  Miller  of  the  National  Board  of  Fire 
Underwriters. 

Conflagrations  at  Seattle,  Spokane,  Jacksonville, 
Waterbury,  Paterson,  Eochester,  etc.,  are  not  taken  into 
account,  as  they  would  figure  with  the  premium  income 
and  population  in  such  group  that  they  might  be  placed. 

Your  Committee  are  of  the  opinion  that  the  public 
having  the  option  of  purchasing  any  percentage  of  in- 


8 

surance  to  value  tliey  wished  would  be  disarmed  from 
making  any  valid  objection  to  co-insurance,  and  the 
companies  would  be  protected  from  adverse  selection 
being  made  against  them  by  having  to  carry  a  high  per- 
centage of  value  on  poor  risks  and  a  small  percentage 
on  good  risks.  With  the  percentage  co-insurance  clause 
and  rates  adjusted  accordingly,  the  business  of  fire 
insurance  will  take  care  of  itself  much  more  evenly  than 
at  present. 

It  is  well  knoAvn  that  in  years  of  general  depres- 
sion in  business  the  loss  ratio  to  premium  income 
increases  as  a  rule;  is  not  that  accounted  for  by  the  fact 
that  property  owners  studying  economy  often  reduce 
the  amount  of  insurance  carried,  and  the  adverse  expe- 
rience of  insurance  companies  in  such  years  can  with 
little  doubt  (if  any),  in  the  Committee's  opinion,  be 
attributed  to  the  falling  off  in  contributing  insurance 
to  partial  losses.  In  other  words,  the  adverse  selection 
by  the  insured  reducing  the  average  amount  of  insurance 
carried  to  value  increases  loss  ratio  and  turns  profit  into 
loss.  This  is  a  much  more  reasonable  solution  of  the 
unprofitable  results  to  insurance  companies  during 
years  of  business  depression  than  the  easier  and  oft- 
reiterated  cry  of  increased  moral  hazard. 

Eepeating  our  hope  that  this  report  may  pave  the  way 
for  Eastern  associations  to  take  this  subject  up  on  a 
similar  or  better  basis  (if  ascertainable)  and  reduce  to 
a  science  the  subject  of  percentage  co-insurance  and  rela- 
tive rates  to  be  charged  therefor. 

Eespectfully  submitted, 

C.   F.   MULLINS, 

Chairman. 
Aethur  M.  Brown, 
r.  w.  osborn, 
B.  J.  Smith, 
V.  Carus  Driffield, 

Committee. 


Mr.  C.  F,  Mullins, 

Chairman  of  the   Co-insurance   Committee   of   the 

Board  of  Fire  Underwriters  of  the  Pacific, 

Dear  Sir:— I  have  the  honor  of  presenting  herewith,  as 

requested  by  your  committee,    a    report   based    on  fire 

statistics  of  the  city  of  San  Francisco  for  the  years  1899- 

1903,  inclusive. 

The  primary  object  of  this  investigation  has  been  the 
determination  for  a  number  of  classes  of  the  relative 
rates  for  co-insurance,  the  secondary  object  has  been  the 
putting  on  record  of  the  elements  involved  in  the  scien- 
tific determination  of  fire  insurance  rates  and  of  a  plan 
for  the  calculations  necessary  thereto.    I  am. 

Yours  very  truly, 

Albert  W.  Whitney. 
Berkeley,  California,  August,  1905. 


11 


I. — Introduction. 

A  determination  of  the  rates  for  co-insurance  involves, 
first,  a  thorough  analysis  of  the  structure  of  a  rate,  sec- 
ond, the  gathering  of  the  necessary  statistics  and,  third, 
the  application  of  the  theory  to  the  facts. 

I  say,  first,  a  thorough  analysis  of  the  rate.  As  a  mat- 
ter of  fact,  co-insurance  rates  are  nothing  but  the  real 
rates  guarded  by  the  co-insurance  clause  against  adverse 
selection,  namely,  the  selection  by  the  insured  of  less  in- 
surance than  the  rate  was  designed  for. 

In  order  to  understand  what,  for  lack  of  a  better  name, 
I  have  called  the  real  rates,  it  is  necessary  to  have  clearly 
in  mind  the  elements  that  make  the  fire  insurance  prob- 
lem; but  in  order  to  understand  the  fire  insurance  prob- 
lem, I  propose,  for  the  sake  of  the  added  clearness  that 
comes  from  comparison,  to  measure  it  against  the  life  in- 
surance problem  whose  elements  are  more  easily  grasped. 

If  a  man  wished  to  buy  an  insurance  of  $1,000  for  his 
whole  life  and  if  the  element  of  interest  were  to  be  neg- 
lected the  price  of  the  insurance  would  be  $1,000,  for  he 
is  sure,  sooner  or  later,  to  die.  The  time  of  his  death  is 
immaterial  when  the  element  of. interest  is  left  out  of 
account,  but  when  this  is  admitted  the  time  of  his  death 
is  a  matter  of  considerable  importance.  If  it  were  known 
positively  that  he  would  live  exactly  twenty  years  and  if 
an  interest  rate  of  five  per  cent  prevailed  the  insurance 
could  be  sold  to  him  for  the  present  value  of  $1,000  twenty 
years  hence,  or  about  $377.  But  there  would  be  no  reason 
for  calling  this  insurance ;  it  would  be  nothing  more  than 
an  ordinary  banking  transaction. 

But  as  a  matter  of  fact  the  time  of  his  death  is  uncer- 
tain, and  it  is  this  element  of  uncertainty  that  brings  the 
transaction  into  the  field  of  insurance.  Yet  all  must  not 
be  uncertainty  or  there  will  be  no  basis  for  an  agreement. 
In  reality  we  may  assume  that  we  know  four  things ;  first, 


12 

a  safe  rate  of  interest  to  count  upon ;  second,  the  age  of 
the  applicant ;  third,  that  he  is  sound  physically  and  has 
a  good  environment ;  fourth,  that  a  large  number  of  men 
of  his  age  and  of  an  average  physical  condition  and  en- 
vironment who  have  in  the  past  been  under  observation 
have  experienced  a  certain  recorded  mortality.  We  may 
treat  this  man  then  as  though  he  were  one  of  such  a 
group.  We  may  thereupon  compute,  taking  account  of 
interest,  the  present  value  of  the  sum  needed  to  meet  the 
death  claims  among  this  group  as  year  by  year  they  ma- 
ture ;  this  sum  assessed  equally  among  the  insured  is  the 
net  single  premium.  As  a  matter  of  fact  life  insurance 
is  usually  paid  for  in  yearly  installments,  but  as  there  is 
no  analogue  to  this  in  fire  insurance  practice,  we  need  not 
follow  it  out. 

I  have  supposed  the  insurance  to  be  for  the  whole  of 
life;  this  eliminates  the  question  of  whether  or  not  the 
claim  will  mature,  and  makes  it  a  question  only  of  when 
it  will  mature. 

This  fact  that  death  is  sure  to  occur,  but  that  a  loss  by 
fire  is  not  sure  to  occur,  has  been  asserted  to  mark  a  vital 
distinction  between  these  two  types  of  insurance.  This  is 
wrong,  however,  for  the  certainty  of  death  does  not  set 
any  characteristic  mark  upon  life  insurance,  and,  as  a 
matter  of  fact,  term  insurance  in  one  form  or  another  is  a 
large  part  of  the  business  of  a  life  insurance  company. 

Term  insurance,  however,  yields  a  problem  even  if  in- 
terest is  neglected.  If  a  man  of  35  wishes  to  insure  his 
life  for  $1,000  during  only  the  next  five  years,  we  may  go 
to  the  mortality  table  and  consider  the  81,822  persons 
alive  at  the  age  of  35 ;  we  shall  find  that  of  this  number 
3,716  die  during  the  next  five  years.  If  each  of  this  group 
of  persons  were  insured,  the  death  claims  would  amount 
to  $3,716,000.  Neglecting  interest  and  assessing  this 
equally  among  the  81,822  persons  insured  would  give  a 
net  single  premium  of  about  $45.     If  interest  is  taken 


13 


into  account  the  problem  is  essentially  the  same  as  the 
problem  of  whole-life  insurance  that  we  have  already 
discussed. 

We  have  assumed  that  we  are  dealing  with  persons  of 
a  fairly  definite  type  and  standard  of  civilization,  for 
instance,  white  persons  living  in  the  northern  part  of  the 
United  States.  If  we  wish  to  transact  an  insurance  busi- 
ness among,  for  instance,  the  natives  of  India,  the  prob- 
lem will  be  the  same  except  that  we  shall  find  a  mortality 
experience  peculiar  to  the  class. 

This  is  the  form  of  a  mortality  table:— 


Table  1.- 

-The  American  Experience  Table 

X 

dx 

10 

749 

11 

74G 

12 

743 

93 

58 

94 

18 

95 

3 

Total,     100,000 

The  column  headed  x  refers  to  the  number  of  completed 
years,  the  column  headed  d^  refers  to  the  number  dying 
during  the  following  year.  The  sum  of  the  numbers  in 
the  second  column  is  100,000;  that  is,  100,000  persons 
began  the  11th  year  together ;  of  these,  the  table  says,  749 
died  during  that  year,  746  during  the  12th  year,  and  so  on. 
Among  the  natives  of  India  we  should  obtain  a  different 
set  of  numbers. 

We  may  now  see  clearly  the  elements  that  go  to  make 
up  a  life  insurance  rate ;  they  are  two,  the  law  of  mortal- 


14 

ity  and  the  rate  of  interest.  The  rate  then  will  depend, 
first,  upon  the  law  of  mortality,  which  will  vary  with  the 
class,  second,  upon  the  age  of  the  insured  at  the  time  at 
which  the  insurance  is  effected,  third,  upon  the  rate  of 
interest,  and,  fourth,  of  course,  upon  the  particular  form 
of  insurance  desired. 

Now  let  us  make  a  corresponding  analysis  of  fire  insur- 
ance conditions.  In  the  first  place  fire  insurance  is 
written  for  such  short  terms  that  the  element  of  interest 
enters  in  such  a  simple  way  as  to  be  negligible  in  making 
the  rate.  We  therefore  sweep  away  at  once  what  is  the 
main  factor  in  the  structure  of  a  life  insurance  rate. 

The  mortality  among  lives  changes  with  the  age.  This 
is  not  true,  however,  among  fire  risks  to  any  great  extent, 
that  is,  the  age  of  a  building  is  not  a  very  important  ele- 
ment in  fixing  the  rate,  at  least  it  need  not  and  indeed 
cannot  be  taken  account  of  in  any  such  systematic  way  as 
in  life  insurance.  This,  together  with  the  fact  that  fire 
insurance  is  written  for  such  short  terms,  eliminates  this 
element  from  the  problem. 

We  see,  therefore,  that  the  two  factors  of  life  insur- 
ance rating,  the  rate  of  interest  and  the  mortality  in 
terms  of  age,  are  almost  lacking  in  the  fire  insurance 
problem,  at  least  they  do  not  require  a  systematic 
treatment. 

What  then  are  the  elements  of  the  fire  insurance 
problem?  In  the  first  place  and  most  important  of  all, 
the  element  of  class.  Just  as  the  rate  for  a  man  with 
an  hereditary  tendency  to  an  organic  disease  or  for  a 
stoker  or  for  an  inhabitant  of  a  tropical  country  should 
be  higher  than  the  rate  for  a  healthy  life  in  a  healthful 
environment,  so  the  rate  for  a  saw-mill  or  a  frame  build- 
ing without  fire  protection  should  be  higher  than  for  a 
fire-proof  office  building.  In  fire  insurance  the  class  is 
more  important  than  any  other  element  in  making  the 
rate,  while  in  life  insurance  it  has  been,  until  recently, 


15 

of  very  little  importance,  for  almost  all  life  insurance 
has  been  effected  in  a  single  class,  that  of  standard 
lives  of  white  persons  in  non-tropical  latitudes. 

We  now  come,  however,  to  the  element  that  really  dif- 
ferentiates fire  insurance  from  life  insurance.  In  life 
insurance  there  is  no  such  thing  as  partial  loss  (the 
analogy,  if  any,  may  be  sought  in  accident  insurance) ; 
when  a  man  dies  the  full  face-value  of  the  policy  is 
drawn  upon.  But  when  a  building  or  a  stock  of  goods 
burns,  it  seldom  burns  completely;  the  amount  of  dam- 
age is  an  exceedingly  important  factor  in  the  problem, 
for  it  determines  what  part  of  the  face  of  the  policy 
will  have  to  be  paid.  This  is  the  analogue  to  the  element 
of  mortality  among  lives,  but  instead  of  the  time  ele- 
ment we  have  the  quantity  element.  In  life  insurance 
the  mortality  question  is  when,  in  fire  insurance  it  is 
how  much.  The  mortality  table  for  life  insurance  gives 
how  many  in  terms  of  when,  the  mortality  table  for  fire 
insurance  gives  how  many  in  terms  of  how  much.  This 
for  instance  is  the  mortality  table  for  frame  business 
buildings  in  San  Francisco: 

*  Table  2.— Table  of    Paktial  Loss  fob  the  Class  of 
Frame  Business  Buildings. 


X 

m^ 

0 

8293 

1 

576 

2 

326 

3 

215 

4 

139 

5 

97 

6 

69 

7 

49 

8 

42 

9 

194 

Total     10,000 


*  It  is  not  to  be  inferred  that  this  table  gives  the  actual  number  of  fires 
that  have  occurred  in  some  particular  time  ;  the  numbers  are  only  relative, 


10,000  having  been  chosen  for  convenience. 


16 

The  column  headed  x  refers  to  the  number  of  tenths 
of  sound  value  next  lower  than  the  amount  destroyed, 
and  the  column  headed  m^,  refers  to  the  number  of  risks, 
among  10,000  losses  altogether,  that  experience  this 
particular  range  of  loss.  That  is,  out  of  10,000  build- 
ings damaged  by  fire  8,293  may  be  expected  to  suffer  a 
loss  of  less  than  1/10  of  the  value,  576  a  loss  of  more 
than  1/10  and  less  than  2/10  of  the  value,  and  so  on.  The 
analogy  of  this  to  a  mortality  table  in  life  insurance  is, 
I  think,  obvious. 

Now  just  as  in  life  insurance  there  are  different  mor- 
tality tables  for  different  classes,  so  in  fire  insurance 
there  are  different  mortality  tables  for  different  classes 
(see  Table  12).  Inside  of  a  single  class  then  the  ele- 
ments of  the  life  insurance  problem  are  the  rate  of  inter- 
est and  the  law  of  mortality  and  the  problem  to  be 
solved  is:  with  a  given  rate  of  interest  and  at  a  given 
age,  what  is  the  rate  for  an  insurance? 

Inside  of  a  single  class  the  element  of  the  fire  insur- 
ance problem  is  the  law  of  mortality,  or  as  I  shall  hence- 
forth call  it,  the  law  of  partial  loss,  and  the  problem  to 
be  solved  is:  with  a  given  ratio  of  insurance  to  value, 
what  is  the  rate  for  an  insurance  t 

As  a  matter  of  fact  when  I  say  that  this  is  the  prob- 
lem of  fire  insurance  I  am  not  stating  the  problem  of 
finding  the  ordinary  rate,  but  the  co-insurance  rate.  The 
ordinary  rate  takes  no  account  of  the  ratio  of  insurance 
to  value,  that  is,  a  man  pays  the  same  rate  for  $5,000 
of  insurance  whether  it  is  written  on  a  building  worth 
$5,500  or  on  one  worth  $10,000.  But  manifestly  the  ex- 
pected loss  to  the  company  is  much  greater  in  the  second 
case  than  in  the  first;  in  the  second  case  it  will  take 
only  a  50  per  cent  loss  to  exhaust  the  insurance,  in  the 
first  case  it  will  take  more  than  a  90  per  cent  loss  to 
exhaust  it.  Now  in  the  case  of  frame  business  buildings, 
according  to  our  table,  there  are  451  chances  of  a  50  per 
cent  loss  to  194  chances  of  a  90  per  cent  loss. 


17 

The  co-insurance  rates  are  special  averages,  the 
ordinary  rate  is  a  general  average  in  which  the  man  who 
insures  for  a  small  amount,  that  is,  a  small  ratio  of 
insurance  to  value,  gets  his  insurance  too  cheaply,  lie 
who  insures  for  a  large  amount  pays  for  his  insurance 
too  dearly. 

The  analogous  rate  in  life  insurance  would  be  obtained 
by  neglecting  the  element  of  age;  then  a  young  man 
would  pay  too  much  for  his  insurance  in  order  that  an 
old  man  might  pay  too  little.  Such  a  rate  would  be  ob- 
tained by  dividing  the  total  amount  of  death  claims  by 
the  total  amount  of  insurance  in  force,  just  as  in  fire 
insurance  the  rate  is  actually  obtained  by  dividing  the 
total  insurance  loss  by  the  insurance  in  force. 

The  weakness  of  a  system  of  this  kind  in  life  insur- 
ance is  this,  that  it  leads  to  adverse  selection;  young 
men  will  not  insure,  therefore  the  mortality  increases, 
the  rate  increases,  the  svstem  is  unstable.  Hence  life 
companies  are  forced  to  take  account  of  the  element  of 
age ;  witness  the  experience  of  friendly  societies. 

We  might  expect  adverse  selection  in  fire  insurance; 
it  would  consist  in  a  refusal  to  carry  much  insurance, 
that  is,  a  large  ratio  of  insurance  to  value.  That  it  does 
not,  as  a  matter  of  fact,  operate  to  a  greater  extent  is 
due  to  several  causes,  one  of  which  is  that  the  insurance 
is  often  needed  at  any  reasonable  price,  especially  as 
collateral  security  for  loans. 

However,  just  as  it  is  manifestly  fairer  and  better 
that  a  man  of  age  25  should  be  rated  according  to  the 
hazard  at  his  age  rather  than  be  forced  to  help  make  up 
the  deficit  caused  by  a  too  small  rate  for  a  man  of  55, 
so  it  is  manifestly  fairer  and  better  that  a  man  who 
wishes  to  insure  for  90  per  cent  of  the  value  of  his  prop- 
erty should  be  given  a  rate  to  meet  the  hazard  rather 
than  be  forced  to  help  make  up  the  deficit  caused  by  the 
under-rating  of  a  man  who  carries  30  per  cent  of  insur- 
ance to  value. 


18 

I  hardly  need  discuss  the  practical  difficulties  in  the 
way  of  this.  One  of  them  arises  from  the  expense  of 
ascertaining  sound  values.  For  the  company  to  deter- 
mine sound  values  accurately  at  the  time  of  effecting 
the  insurance  would  be  practically  out  of  the  question, 
and  in  the  case  of  stocks  of  merchandise  which  are  con- 
tinually changing  would  be  ineffective. 

The  co-insurance  clause  is  an  agreement  on  the  part 
of  the  insured  to  maintain  a  specified  ratio  of  insurance 
to  value.  He  will  maintain  this  in  insurance  companies 
presumably,  but  in  case  he  fails  so  to  do  he  shall  by 
the  agreement  be  regarded  as  himself  a  co-insurer  for 
the  balance.  He  thereby  becomes  jointly  responsible 
with  the  other  insurers,  each  for  his  share  of  the  loss. 
This  agreement  places  upon  the  insured  the  responsi- 
bility for  the  ascertaining  of  sound  values  and  with 
entire  fairness  for  he  should  have  a  sufficiently  accurate 
knowledge  of  his  own  affairs  to  obtain  this  information 
easily  and  to  order  his  business  with  this  agreement  in 
mind. 

To  fix  the  proper  rate  for  a  fire  insurance  it  is  as 
necessary  to  have  this  information  as  to  ratio  of  insur- 
ance to  value  as  to  know  the  age  of  an  applicant  for 
life  insurance.  The  responsibility  for  stating  his  age 
correctly  is  placed  upon  the  insured  with  a  penalty  if 
he  fails  to  do  so;  the  co-insurance  clause  puts  upon  the 
insured  the  responsibility  for  keeping  the  ratio  of 
insurance  to  value  at  a  specified  figure  with  a  penalty 
if  he  fails  to  do  so  of  having  to  act  himself  as  insurer 
for  the  balance. 

Or  again:  when  a  man  buys  $8,000  worth  .of  insur- 
ance and  pays  for  it  at  an  80  per  cent  co-insurance  rate 
it  is  equivalent  to  the  admission  that  his  insurance 
protects  just  $10,000  worth  of  property  (of  course  only 
partially  protects,  namely,  up  to  80  per  cent  of  its 
value).    The  fact  that  he  buys  protection  for  just  this 


19 

amount  of  property  is  an  integral  part  of  the  transac- 
tion. If  when  a  loss  occurs  the  sound  value  of  the 
property  is  greater  than  $10,000  it  is  quite  obvious  that 
the  excess  value  is  in  this  sense  unprotected,  or,  if  you 
like,  is  insured  by  himself  for  80  per  cent  of  its  value. 
In  case  the  damage  is  less  than  80  per  cent  the  insured 
bears  of  the  loss  only  his  part  as  a  co-insurer;  if  the 
loss  is  greater  than  80  per  cent  he  loses  not  only  as  a 
co-insurer,  but  also  because  he  has  bought  only  incom- 
plete protection. 

The  rates  for  co-insurance  then  are  nothing  but  rates 
that  take  into  account  ratio  of  insurance  to  value.  The 
necessary  and  sufficient  data  for  their  determination  for 
a  particular  class  are  contained  in  what  I  have  called 
the  table  of  partial  loss.     . 

The  mathematical  statement  of  the  method  actually 
used  in  computing  these  rates  will  be  reserved  till  later, 
for  it  is  not  necessary  to  a  general  understanding  of  the 
subject.  I  shall  defer  also  the  explanation  of  the  method 
of  obtaining  this  table  of  partial  loss  from  the  office 
statistics. 

II.— The  Co-Insurance  Problem  Treated  Arith- 
metically. 

As  an  example,  let  us  consider  the  60  per  cent  co- 
insurance* rate  on  frame  business  buildings.  The  table 
of  partial  loss  has  already  been  given  (Table  2).  As  a 
rough  approximation  we  might  call  the  average  loss 
under  10  per  cent,  5  per  cent,  the  average  loss  between 
10  and  20  per  cent,  15  per  cent,  and  so  on,  but  as  a  matter 
of  fact  it  is  worth  while  to  examine  our  statistics  closely 
enough  to  determine  more  accurate  averages.  The 
results  are  as  follows: 

*  Whenever  rale  is  referred   to  in  this  report,   net  rate  or  fire  cost  is 
meant.    The  ofl&ce  rate  is  obtained  from  this  simply  by  loading. 


20 


Table   3.— Average   Percentage   of   Property   Loss   to 

Sound  Value. 


1.8  per  cent,  among-  losses  between    0  per  cent. 

and     10  per  c 

14.2      ''           "         ''         ''        10 

'^      20      " 

24.5      ''           ''         "         ''        20 

CC         3Q          .. 

34.7      ''           ''         ''         "        30 

''      40      " 

44.8      ''           ''         ''         ''        40 

''      50      '' 

54.9      ^'           "         "         "        50 

''      60      '' 

65         "           "         ''         "        60 

''      70      '' 

75         ''          ''         '^         ''        70 

''      80      '' 

85         ''           ''         "         "        80 

''      90      '' 

99.5      "           ''         ''         ''        90 

''    100      '' 

cent. 


Let  us  now  find  the  insurance  loss  on  10,000  claims, 
supposing  that  the  sound  value  of  each  risk  is  $100  and 
that  on  each  risk  an  insurance  of  60  per  cent  of  the  value, 
that  is,  $60,  is  carried.  But  before  we  do  this  let  us  find 
the  property  loss.  This  will  evidently  be  made  up  as  fol- 
lows: 


Table  4.  — The  Property  Loss;  Sound  Value  of  Each 
EiSK  $100;  10,000  Claims. 

8293  losses  of  $  1.80  each .$14,927  40 

8,179  20 

7,987  00 

7,460  50 

6,227  20 

5,325  30 

4,485  00 

3,675  00 

3,570  00 

19,303  00 


576 

14.20     '' 

326 

24.50    '' 

215 

34.70    ^' 

139 

44.80     '' 

97 

54.90     '' 

69 

65.00    '' 

49 

75.00    '' 

42 

85.00     '' 

194 

99.50    '' 

Entire  property  loss $81,139  60 

Now  the  insurance  loss ;  since  there  is  an  insurance 
of  $60  on  each  risk,  any  loss  under  $60  will  be  paid  in 
full,  but  for  losses  over  $60  only  $60  on  each.  The  in- 
surance loss  will,  therefore,  be: 


21 


Table  5.— The  Insurance  Loss;  Sound  Value  of  Each 
Risk  $100 ;  Insurance  $60;  10,000  Claims. 

8293  losses  of  $  1.80  each $14,927  40 

8,179  20 

7,987  00 

7,460  50 

6,227  20 

5,325  30 

4,140  00 

2,940  00 

2,520  00 

11,640  00 


576 

14.20  '' 

326 

24.50  '' 

215 

34.70  '^ 

139 

44.80  '' 

97 

54.90  '' 

69 

60.00  '' 

49 

60.00  '' 

42 

60.00  " 

194 

60.00  " 

Insurance  loss $71,346  60 

Now  if  we  divide  this  insurance  loss  by  the  number 
of  risks  of  this  kind  among  which  these  10,000  losses 
have  occurred,  we  shall  obtain  the  average  insurance 
loss  per  risk.  This  will  be  the  average  insurance  loss  per 
risk  for  $60  of  insurance ;  the  average  insurance  loss  per 
risk  per  dollar  of  insurance  will  be  had  by  dividing  this 
by  60.  This  will  be  then  the  net  rate  or  fire  cost  per 
dollar  of  insurance  when  the  insurance  carried  is  60  per 
cent  of  the  sound  value.  But  as  a  matter  of  fact  the  num- 
ber of  risks  upon  which  these  10,000  losses  have  occurred 
was  not  obtainable  from  the  statistics  at  hand  and 
therefore  it  has  been  impossible  to  determine  the  actual 
rate. 

But  relative  rates  are  easily  enough  determined;  for 
just  as  the  insurance  loss  has  been  determined  for  a 
ratio  of  insurance  to  value  of  60  per  cent,  so  the  insur- 
ance loss  may  be  determined  for  any  percentage  of 
insurance  to  value.     The  results  are  as  follows: 


22 


Table    6.— Insurance    Losses    for    Various    Eatios    of 
Insurance  to  Value;  Sound  Value  of  Each 
EiSK  $100;  10,000  Claims. 


ratio  op  insurance  to  value. 


10  per 
20 
30 
40 
50 
60 
70 
80 
90 
100 


cent, 


THE  insurance  LOSls. 


,$31,997  40 
,  45,726  60 
,   55,243  60 

62,154  10 
,  67,331  30 
,  71,346  60 
,  74,541  60 
,  77,146  60 

79,296  60 
,  81,139  60 


With  full  insurance  the  insurance  loss  is  equal  to  the 
property  loss   as  it  should  be. 

Now  just  as  we  proposed  to  obtain  the  60  per  cent 
rate  by  dividing  first  by  the  number  of  risks  and  then 
by  60  so  we  might  propose  to  obtain  the  10  per  cent 
rate  by  dividing  by  the  number  of  risks  and  then  by  10, 
and  so  on.  Now  since  the  unknown  element,  the  number 
of  risks,  enters  into  all  the  rates  in  the  same  way,  in 
forming  the  relative  rates  it  may  be  neglected  and  we 
may  divide  the  insurance  losses  in  Table  6  simply  by 
the  corresponding  amounts  of  insurance  carried.  This 
gives: 


23 

Table  7.— Relative  Rates  for  Various  Ratios  of  Insur- 
ance TO  Value. 


ratio  of  insurance  to  value. 


10  per 
20 
30 
40 
50 
60 
70 
80 
90 
100 


cent, 


relative  rates. 


,3200 
,2286 
.1841 
.1554 
.1347 
.1189 
.1065 
.  964 
.  881 
.  811 


From  Table  6  we  have 
insurance  losses : 


by   subtraction  of   successive 


Table  8.— Cost  to  the  Company  of  Carrying  Successive 

Tenths  of  Insurance  to  Value  ;  Sound  Value 

OF  Each  Risk  $100;  10,000  Claims. 

The     1st  tenth $31,997  40 


2d 
3d 
4th 
5th 
6th 
7th 
8th 
9th 
10th 


13,729  20 
9,517  00 
6,910  50 
5,177  20 
4,015  30 
3,195  00 
2,605  00 
2,150  00 
1,843  00 


It  is  hardly  necesary  to  point  out  how  severely  the 
first  few  tenths  of  insurance  carried  tax  the  company 
in  comparison  with  the  later  tenths.  The  cost  to  the 
company  of  carrying  an  insurance  of  20  per  cent  of  the 
value  is  more  than  half  the  cost  of  carrying  full  insur- 
ance (see  Table  6).  The  rate  for  10  per  cent  insurance 
is  four  times  the  rate  for  full  insurance  (see  Table  7). 


24 


The  question  that  now  comes  up  to  be  answered  is 
this:  what  relation  has  the  ordinary  rate  to  the  co- 
insurance rates'?  This  cannot  be  answered  without 
additional  statistical  information.  The  information  that 
we  need  is  this:  how  much  insurance  do  people  buy  in 
general?  If  on  the  average  they  insure  their  buildings 
for  20  to  30  per  cent  of  the  value  the  rate  will  be  high, 
if  on  the  other  hand  they  insure  well  up,  on  the  aver- 
age say  for  80  per  cent  of  the  value,  the  rate  will  be  low. 
Roughly  we  might  say  that  the  ordinary  rate  will  be  the 
same  as  the  co-insurance  rate  for  the  average  ratio  of 
insurance  to  value.  This,  however,  in  general  will  not 
be  exactly  true  and  we  have  at  hand  data  that  will  allow 
us  to  make  a  closer  determination.  The  same  statistics 
that  yield  the  table  of  partial  loss  yield  also  data  regard- 
ing amount  of  insurance  carried. 

Out  of  127  risks  under  observation  in  the  class  of 
frame  business  buildings  under  discussion,  two  carried 
between  20  and  30  per  cent  of  insurance,  five  carried 
between  30  and  40  per  cent,  or  in  tabular  form : 

*  Table  9.  — The  Distribution  of  Risks  as  Regards  Ratio 
OF  Insurance  to  Value. 


ratio  of  insurance  to  value. 


Between    0  per  cent  and  10  per  cent 
10       '*         ''     20       '' 


20 
30 
40 
50 
60 
70 
80 
Over  90  per  cent 


20 
30 
40 
50 
60 
70 
80 
90 


Total 127 


number  of  risks. 


0 
0 
2 
5 

14 
14 
29 
21 
22 
20 


*  I  shall  refer  to  this  as  the  table  or  law  of  insurance  to  value. 


25 

The  average  ratio  of  insurance  to  value  taken  from 
the  actual  figures  was  70.77  per  cent. 

If  we  were  to  divide  each  of  the  insurance  losses  in 
Table  6  by  the  number  of  risks  we  should  obtain  the 
average  insurance  losses  per  risk,  but  as  I  have  said 
we  have  no  information  as  to  the  number  of  risks.  How- 
ever, if  we  divide  each  of  these  losses  by  the  number 
of  risks  upon  which  loss  occurred,  that  is  10,000,  we 
should  obtain  the  average  insurance  loss  per  claim.  For 
instance  the  average  insurance  loss  per  claim  when  60 
per  cent  of  insurance  to  value  is  carried  is  $7.13466; 
when  70  per  cent  is  carried  is  $7.45416,  and  for  risks 
that  range  in  ratio  of  insurance  to  value  from  60  to  70 
per  cent  as  do  the  29  in  Table  9  we  may  with  sufficient 
accuracy  take  for  the  average  insurance  loss  per  claim 
the  average  of  the  60  and  the  70  per  cent  values  or 
$7,294  and  call  this  the  average  insurance  loss  per  claim 
when  65  per  cent  of  insurance  to  value  is  carried,  and 
so  for  the  other  intervals. 

These  intermediate  values  of  the  average  insurance 
loss  per  claim  are : 

Table  10.— The  Average  Insurance  Loss  per  Claim  for 

Various  Intermediate  Values  of  the  Eatio  op 

Insurance  to  Sound  Value. 


ratio  of  insurance  to 

VALUE. 


5  per  cent, 
15 
25 
35 
45 
55 
65 
75 
85 
95 


a 

a 

i  i 

i  i 
n 


AVERAGE  INSURANCE  LOSS 
PER  CLAIM. 

$1,600 

3.886 

5.048 

5.869 

6.474 

6.934 

7.295 

7.584 

7.822 

8.022 


26 


Now  then,  assuming  that  the  experience  in  Table  9 
may  be  taken  to  be  typical,  2/127  of  10,000  claims,  on 
which  the  insurance  averages  25  per  cent  of  the  value,  will 
have  an  average  insurance  loss  per  claim  of  $5,048,  or 
altogether  $795.00 ;  5/127  of  10,000  claims,  on  which  the 
insurance  averages  35  per  cent  of  the  value,  will  have  an 
average  insurance  loss  per  claim  of  $5,869,  or  altogether 
$2311.00 ;  and  so  on,  or  in  tabular  form : 

Table    11.  — The    Actual   Insurance   Loss     on     10,000 

Claims;  Sound  Value  of  Each  Risk  $100;  Amount 

OF  Insurance  Carried  as  in  Table  9. 


AVERAGE  ratio  OF  INSURANCE  TO 
VALUE. 


5 

15 
25 
35 
45 
55 
65 
75 
85 
95 


per  cent, 


actual  insurance 
loss. 


,$  795  00 
.  2,311  00 
.  7,137  00 
.  7,644  00 
.  16,656  00 
12,540  00 
13,550  00 
.   12,633  00 


Actual  insurance  loss  on  10,000  claims $73,266  00 

Dividing  this  by  10,000  we  obtain  $7.3266  as  the  actual 
average  insurance  loss  per  claim;  I  say  actual  since  this 
is  based  upon  figures  as  to  insurance  actually  sold. 

$7.3266  then  is  the  actual  average  insurance  loss  per 
claim,  but  there  is  an  average  insurance  per  claim  of 
70.77  per  cent  of  the  value,  or  70.77  dollars.  The  rate 
then  per  dollar  of  insurance  actually  in  force  will  be 
$7.3266  divided  by  70.77,  or  .1035.  This  is  the  rate  per 
dollar  of  insurance  actually  in  force  on  risks  that  have 
become  claims.  If  this  were  multiplied  by  the  ratio  of 
the  number  of  claims  to  the  number  of  risks    it  would 


27 

be  the  burning  ratio  or  net  rate.  The  corresponding 
rate  (see  Table  7)  with  70  per  cent  co-insurrance  is  .1065, 
with  80  per  cent  co-insurance  is  .0964;  this  actual  rate 
of  .1035  lies  between  these  two  and  by  interpolation  it  is 
found  to  agree  with  the  rate  for  73  per  cent  co-insurance. 
The  corresponding  problem  in  life  insurance  would  be 
to  determine  the  age  for  which  the  rate  would  be  the 
same  as  the  rate  irrespective  of  age  got  by  dividing  the 
total  amount  of  death  claims  by  the  total  amount  of 
insurance  in  force. 

If  then  the  rate  in  use  has  been  properly  obtained, 
that  is,  if  it  is  the  true  rate,  then  if  it  is  taken  for  the  73 
per  cent  co-insurance  rate  and  the  other  co-insurance 
rates  are  taken  in  proper  ratios  to  this  as  determined 
by  Table  7,  the  business  will  produce  with  these  rates 
an  amount  sufficient  to  meet  the  expected  loss  just  as 
well  as  with  the  old  single  rate;  in  fact  even  if  the  old 
rate  is  not  the  true  rate  but  if  it  is,  for  instance,  too 
large,  the  business  conducted  with  the  co-insurance  rates 
will  continue  to  produce  the  same  income  as  with  the 
old  single  rate,  provided  the  law  of  insurance  to  value 
remains  the  same. 

A  very  important  thing  to  observe  is  that  the  co-insur- 
ance rates  are  entirely  independent  of  the  law  of  insur- 
ance to  value;  that  was  introduced  only  when  we  came 
to  a  consideration  of  the  ordinary  rate.  As  a  matter 
of  fact  the  tendency  would  be  with  a  change  to  co-insur- 
ance rates  for  the  average  amount  of  insurance  carried 
to  increase.  The  co-insurance  rates,  however,  would 
still  continue  perfectly  to  produce  the  income  requisite 
to  meet  the  expected  loss.  The  expected  loss  would  now 
be  larger  but  it  would  not  increase  as  fast  as  the  amount 
of  insurance  in  force  so  that  the  ratio  of  the  two  or  the 
ordinary  rate  would  fall.  As  the  profit  is  proportional 
to  the  expected  loss  the  ratio  of  profit  to  insurance  in 
force    will    be    proportional   to   the   ordinary   rate;   a 


28 

change  then  to  co-insurance  rates  would  be  likely  to  be 
followed  by  an  increase  in  the  profit,  but  not  as  great 
an  increase  as  that  in  the  insurance  in  force. 

While  then  the  co-insurance  rates  have  the  great 
advantage  of  being  independent  of  the  law  of  insurance 
to  value  so  that  in  any  case  the  business  will  take  care 
of  itself  so  long  as  the  law  of  partial  loss  does  not  change, 
the  ordinary  rate  on  the  other  hand  depends  very  decid- 
edly on  the  law  of  insurance  to  value  so  that  with  this 
rate  the  business  would  not  take  care  of  itself  if  there 
were  a  change  in  this  law;  for  instance  a  drop  in  the 
average  ratio  of  insurance  to  value  from  say  70  to  50 
per  cent  would  probably  convert  a  profitable  business 
into  a  losing  business. 

Before  proceeding  to  give  tabular  results  for  the  sev- 
eral classes  examined,  I  propose  to  state  once  more  the 
elements  involved  in  a  rate  and  to  suggest  some  terms. 

The  basis  on  which  the  computation  is  made  is  what 
I  have  called  the  table  of  partial  loss.  This  will  vary 
from  one  class  to  another.  The  table  lacks,  however,  one 
element  of  being  competent  to  give  us  absolute  rates; 
it  tells  us  how  the  claims  are  distributed  as  to  size,  but 
it  does  not  tell  us  what  proportion  of  risks  become  claims. 
The  table,  however,  is  competent  to  give  us  relative 
rates  and  to  give  us  the  relation  between  the  co-insur- 
ance rates  and  the  ordinary  rate. 

Let  us  suppose  for  a  moment  that  we  had  this  addi- 
tional information.  Then  we  should  be  able  to  compute 
the  insurance  loss  for  any  given  percentage  of  insurance 
to  value,  not  merely  per  claim,  but  per  risk,  and  from  this 
per  dollar  of  insurance.  This  would  be  the  net  co-insur- 
ance rate,  the  fire  cost  or,  I  should  like  to  call  it,  the  meas- 
ure of  the  hazard.  We  should  find  on  examination  that 
this  consists  of  two  factors,  one  being  the  ratio  of  the 
number  of  claims  to  the  number  of  risks ;  this  can  be  in- 
terpreted as  the  probability  that  a  fire  will  occur;  this  I 


29 

propose  to  call  the  ignition  hazard.  The  second  factor  is 
the  probability  that,  a  fire  having  started,  the  amount  at 
stake,  namely,  the  amount  of  insurance  on  the  risk,  will 
be  lost  to  the  company ;  this  I  propose  to  call  the  damage 
hazard.  The  product  of  these  two,  the  ignition  hazard 
and  the  damage  hazard,  is  the  measure  of  the  hazard  or 
the  fire  cost  for  this  particular  ratio  of  insurance  to  value ; 
it  is  the  probability  first  that  a  fire  will  occur,  and,  sec- 
ond, that  having  occurred,  there  will  be  a  loss  of  the 
insurance.* 

The  ignition  hazard  might  be  the  same  under  some  con- 
ditions on  a  stock  of  millinery  in  a  brick  building  as  on  a 
stock  of  groceries,  but  the  damage  hazard  in  the  case  of 
the  millinery  would  certainly  be  much  larger  than  in  the 
case  of  the  groceries. 

The  ignition  hazard  is  independent  of  the  ratio  of  in- 
surance to  value,  the  damage  hazard  on  the  other  hand 
depends  upon  this  as  well  as  upon  the  susceptibility  to 
damage  of  the  risk  since  it  is  the  probability  of  a  loss  of 
the  insurance. 

Just  as  we  have  analyzed  the  co-insurance  rates  so  we 
may  analyze  the  ordinary  rate.  It  will  be  found  to  con- 
sist in  the  same  way  of  two  factors,  the  ignition  hazard 
and  the  damage  hazard.  The  ignition  hazard,  since  it  de- 
pends only  upon  the  class,  will  have  exactly  the  same 
value  as  in  the  co-insurance  rates;  the  damage  hazard, 
which  is  the  probability  that  the  amount  at  stake,  namely, 
the  average  amount  of  insurance  per  risk,  will  be  a  loss 
to  the  company,  will,  however,  differ  from  the  damage 
hazards  with  co-insurance. 

*  It  should  be  noted  that  in  reality  the  insurance  loss  per  risk  from  which 
the  fire  cost  is  got  is  a  very  complex  thing  made  up  as  it  is  of  the  sum  of  a 
number  of  partial  losses,  some  of  which  do  and  some  of  which  do  not  ex- 
haust the  insurance  It  however  reduces  to  the  form  of  a  single  simple  ex- 
pectation of  losing  the  whole  amount  at  stake,  so  that  for  instance  in  the 
case  of  60  per  cent  co-insurance  already  discussed,  the  insurance  loss  per 
claim  made  up  of  separate  items  as  in  Table  5  (dividing  by  10000)  reduces  to 
a  single  quantity  17.1346  which  is  the  amount  at  stake,  |60,  multiplied 
by  ,1189,  the  damage-hazard.  The  fire-cost  comes  by  multiplying  this  by  the 
ignition-hazard. 


30 

The  ignition  hazard  involves  the  very  element  that  our 
statistics  fail  to  give  and  is  therefore  treated  as  unknown 
in  this  report.  The  damage  hazards,  on  the  other  hand, 
we  can  compute  completely.  As  the  practical  problem  of 
relative  rates  for  co-insurance  involves  only  the  damage 
hazards  it  is  therefore  entirely  soluble. 

III.— Eesults  foe  Eight  Classes  in  Tabular  Form. 

It  was  originally  intended  to  embrace  in  the  investiga- 
tion ten  classes,  first,  frame  business  buildings;  second, 
contents  of  the  same;  third,  brick  business  buildings; 
fourth,  contents  of  the  same;  fifth,  dwellings,  (frame); 
sixth,  contents  of  the  same;  seventh,  frame  special  haz- 
ards; eighth,  contents  of  the  same;  ninth,  brick  special 
hazards ;  tenth,  contents  of  the  same.  It  was  found,  how- 
ever, upon  examination  of  the  statistics  that  they  were  in- 
sufficient in  number  in  the  last  two  classes  to  give  reliable 
results.  The  results  for  the  other  classes  are  given  here- 
with in  tabular  form:' 


31 


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haiard,  that  is  at  which  the  co-insurance  rate  is 
equal  to  the  ordinary  rate. 

36 


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37 

IV.— The  Statistical  Pkoblem. 

In  the  foregoing  pages  I  have  tried  to  show  that  the  co- 
insurance problem  may  be  solved  when  the  laws  of  partial 
loss  and  of  ratio  of  insurance  to  value  are  known,  but  as 
a  matter  of  fact  by  far  the  most  laborious  part  of  this 
investigation  has  been  the  determination  of  these  laws, 
even  taking  no  account  therein  of  the  difficulty  of  getting 
the  statistics. 

No  data  in  the  least  adequate  for  such  an  investigation 
as  this  were  immediately  at  hand  and  there  was  no  possi- 
bility of  obtaining  them  except  from  proofs  of  loss  in  the 
separate  offices.  A  letter  addressed  to  the  Board  offices 
by  the  Chairman  of  the  Co-insurance  Committee  request- 
ing that  access  to  proofs  of  loss  for  the  years  1899-1903, 
inclusive,  be  given  to  myself  or  a  representative,  brought 
a  favorable  response  from  about  ninety  companies. 

The  work  of  examining  this  material  and  collecting 
from  it  the  necessary  data  was  excellently  done  by  Mr.  A. 
H.  Mowbray,  now  in  the  Actuarial  Department  of  the 
New  York  Life  Insurance  Company;  Mr.  Mowbray  was 
assisted  for  a  time  by  Mr.  Hart  Greensfelder. 

The  card  system  was  used;  each  card  represented  a 
risk  upon  which  a  fire  loss  had  occurred;  on  each  card 
were  places  for  the  date  and  location  of  the  fire  and 
marks  for  avoiding  and  detecting  the  multiple  recording 
of  a  risk  which  was  insured  in  more  than  one  company. 

The  information  really  desired  was  the  class  to  which 
the  risk  belonged,  the  sound  value,  the  amount  of  insur- 
ance carried  and  the  value  of  the  property  destroyed; 
there  were  places  for  these  data  on  each  card. 

If  it  had  been  possible  to  obtain  information  on  every 
risk  with  regard  to  each  of  these  four  items  the  problem 
and  its  treatment  would  have  been  comparatively  simple. 
Three  of  the  items,  the  class,  the  property  loss  and  the 
amount  of  insurance  carried  were  indeed  obtainable  on 
practically  every  risk,  but  the  fourth  item,  the  sound 
value,  in  only  about  twenty  per  cent  of  the  cases. 


38 

We  have  seen  that  the  co-insurance  problem  arises  en- 
tirely from  the  circumstance  of  varying  ratios  of  loss  to 
sound  value.  To  have  information  as  to  the  sound  value 
is  then  absolutely  essential  for  the  solution  of  the  problem 
and  yet  the  situation  was  this,  that  in  eighty  per  cent  of 
the  cases  the  sound  value  was  not  obtainable. 

This  was  so  discouraging  as  to  make  it  seem  almost  im- 
possible to  succeed  with  the  investigation.  The  ray  of 
hope  that  came  to  one  for  an  instant  of  being  able  to 
throw  aside  the  eighty  per  cent  of  cases  where  sound 
>alue  was  not  given  and  treat  only  the  remaining  cases 
disappeared  completely  as  soon  as  one  realized  that  the 
cases  in  which  sound  value  is  given  are  highly  selective, 
they  are,  namely,  in  general,  or  at  least  largely,  just  those 
cases  in  which  the  loss  has  been  relatively  large,  and  they 
would  therefore  yield  entirely  misleading  results. 

The  question  then  is :  how  can  we  introduce  the  element 
of  sound  value  into  this  eighty  per  cent  of  cases  in  which 
we  have  only  property  loss  and  amount  of  insurance 
given  1 

Fortunately  this  is  not  quite  so  hopeless  as  it  seems  at 
first.  If  every  risk  were  insured  for  just  three-quarters 
of  its  value  we  should  be  able  to  infer  the  value  from  the 
amount  of  the  insurance  and  therefore  the  ratio  of  loss  to 
value,  a  fifty  per  cent  ratio  of  loss  to  insurance,  for  ex- 
ample, would  mean  a  37i/^  per  cent  ratio  of  loss  to  value. 

Now  it  can  indeed  be  shown  that  the  average  ratio  of 
insurance  to  value  is  somewhere  near  75  per  cent  and  it  is 
evident  that  a  fairly  good  approximate  result  would  be 
obtained  if  we  assumed  that  75  per  cent  of  insurance  (or 
whatever  the  exact  value  of  the  average  might  be)  were 
carried  on  each  risk,  relying  upon  the  risks  carrying  over 
75  per  cent  to  offset  those  carrying  less  than  75  per  cent. 
But  as  a  matter  of  fact  this  offsetting  would  by  no  means 
be  perfectly  effective.  Suppose,  for  instance,  the  ratio  of 
insurance  to  value  in  one  case  to  be  120  per  cent  (as  might 


39 

easily  happen  on  stocks  of  goods)  and  in  another  case  to 
be  30  per  cent,  an  average  of  75  per  cent,  and  suppose 
that  the  ratio  of  loss  to  insurance  has  been  in  each  case 
5/6 ;  then,  as  a  matter  of  fact,  on  one  the  ratio  of  loss  to 
value  would  be  120  per  cent  of  5/6,  that  is  a  total  loss,  in 
the  other  case  a  loss  of  only  30  per  cent  of  5/6,  or  one- 
fourth  of  the  value.  Now  if  the  average  ratio  of  insur- 
ance to  value,  75  per  cent,  were  used  we  should  obtain  two 
losses  of  75  per  cent  of  5/6,  or  62i/^  per  cent  of  the  value. 
But  from  the  point  of  view  of  co-insurance  two  losses  of 
621/2  per  cent  of  the  value  are  very  different  from  one 
total  loss  and  another  loss  of  25  per  cent.  This  is  a  case 
where,  in  forming  an  average,  we  lose  the  very  facts  that 
we  need,  that  is  this  is  a  problem  that  needs  something 
finer  than  an  ordinary  average. 

It  is  to  be  said,  however,  that  75  per  cent  (or  whatever 
it  may  more  exactly  be)  is  not  only  the  average  ratio  of 
insurance  to  value  but,  as  a  matter  of  fact,  the  greater 
part  of  all  risks  are  written  for  nearly  this  ratio  and  for 
these  the  use  of  the  average  ratio  instead  of  the  actual 
ratios  would  lead  to  sufficiently  accurate  conclusions. 
The  cases  in  which  we  should  be  led  into  error  are,  how- 
ever, of  considerable  importance.  We  should  lose  for  one 
thing  nearly  all  cases  of  total  loss,  a  fully  insured  total 
loss,  for  instance,  would  count  only  as  a  75  per  cent  loss. 
There  is,  however,  again  this  to  be  said  that  in  nearly  all 
cases  where  there  have  been  relatively  large  losses  the 
sound  values  are  given  and  we  should,  therefore,  not  be 
driven  to  the  expedients  that  we  are  discussing. 

However,  the  point  is  simply  this,  that  as  a  matter  of 
fact  there  is  a  better  method  of  procedure  available  than 
that  involved  in  assuming  a  uniform  75  per  cent  ratio  of 
insurance  to  value  and  that  the  importance  of  the  prob- 
lem demands  its  use. 

I  will  explain  the  method  actually  used  by  applying  it 
to  the  class  of  frame  business  buildings.    In  this  class  our 


40 


statistics  furnished  records,  during  the  five  years,  of  567 
losses;  for  127  of  these,  or  about  22  per  cent,  the  sound 
value  was  given.  For  these  127  risks  for  which  sound 
value  was  given  the  table  of  partial  loss  and  the  table  of 
ratio  of  insurance  to  value  were  as  follows : 


Table  18,  Partial  Loss.    Table  19,  Insurance  to  Value. 


X 

IDx 

0 

71 

1 

17 

2 

12 

3 

6 

4 

4 

5 

4 

6 

2 

7 

1 

8 

2 

9 

8 

Total,     127 


X 


n. 


0 

0 

1 

0 

2 

2 

3 

5 

4 

14 

5 

14 

6 

29 

7 

21 

8 

22 

9 

13 

10 

4 

11 

1 

12 

1 

13 

0 

14 

1 

Total, 

127 

There  is,  furthermore,  a  table  of  ratios  of  loss  to  in- 
surance starting  out  with  64  risks  for  which  this  ratio  is 
less  than  10  per  cent,  14  for  which  the  ratio  is  greater 
tlian  10  and  less  than  20  pev  cent,  and  so  on.  Such  a  table 
as  this  contains  all  the  information  to  be  had  on  such  risks 
as  the  440  upon  which  sound  value  is  not  given. 

However,  even  these  three  tables  do  not  exhibit  all  the 
information  obtainable  from  our  statistics;  we  exhibit, 
for  instance,  17  losses  of  between  10  and  20  per  cent  of  the 
value,  but  we  do  not  exhibit  just  what  per  cent  of  insur- 
ance is  carried  on  each  of  these  particular  losses.  To 
exhibit  our  information  completely  we  must  use  a  double 
entry  table  such  as  the  following: 


41 


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42 

Distances  measured  to  the  right  refer  to  ratio  of  loss 
to  insurance,  distances  up  refer  to  ratio  of  insurance  to 
value,  starting  in  each  case  at  the  lower  left-hand  corner 
at  zero  and  increasing  by  10  per  cent  intervals ;  the  fig- 
ures refer  to  the  number  of  risks  having  at  the  same  time 
the  indicated  values  of  these  two  quantities,  for  instance 
there  are  three  risks  on  which  the  insurance  was  between 
80  and  90  per  cent  of  the  value,  and  on  which  the  loss  was 
between  20  and  30  per  cent  of  the  insurance.  The  figures 
of  Table  19  and  the  table  of  ratio  of  loss  to  insurance  re- 
ferred to  are  found  by  summing  this  table  horizontally 
and  vertically ;  for  instance  without  regard  to  the  ratio  of 
loss  to  insurance  we  find  29  risks  for  which  the  ratio  of 
insurance  to  value  is  between  60  and  70  per  cent. 

Now  if  our  proposed  assumption  of  75  per  cent  ratio  of 
insurance  to  value  were  really  true  we  should  have  no 
such  vertical  distribution  as  this,  but  the  figures  would  be 
massed  in  the  eighth  row. 

We  are  able  to  construct  such  a  table  as  this  because 
we  have  sound  values  given.  If  sound  values  were  not 
given  our  only  knowledge  would  be  that  of  ratio  of  loss  to 
insurance,  that  is  we  should  know  the  numbers  that  cor- 
respond to  those  that  sum  the  columns,  but  we  should 
not  know  them  in  the  distributed  form.  If,  however,  we 
could  only  distribute  these  numbers  in  such  manner  as 
the  numbers  in  the  table  are  distributed  we  should  be  in  a 
fair  way  to  solve  our  problem,  for,  if  we  know  both  the 
ratio  of  insurance  to  value  and  the  ratio  of  loss  to  insur- 
ance, by  multiplication  we  obtain  the  ratio  of  loss  to 
value,  the  desired  information.  To  make  this  more  defi- 
nite, consider  the  actual  figures.  I  have  arranged  the 
440  risks  for  which  sound  value  was  not  known,  grouped 
according  to  ratio  of  loss  to  insurance,  under  the  cor- 
responding groups  in  Table  20. 

Now  although  we  know  that  of  these  440  losses,  380,  for 
instance,  were  for  less  than  10  per  cent  of  the  value,  we 


43 

do  not  know  anything  about  the  ratio  of  the  insurance  to 
the  value.  Is  it  not  reasonable,  however,  to  suppose  that 
the  relative  distribution  as  to  ratio  of  insurance  to  value 
of  these  380  was  much  the  same  as  for  the  corresponding 
64  for  which  sound  value  was  known?  If  then  we  make 
this  assumption,  which  is,  in  short,  that  the  risks  upon 
which  sound  value  is  known  do  not  differ  materially  so  far 
as  their  relative  distribution  as  to  ratio  of  insurance  to 
value  is  concerned  from  the  risks  upon  which  sound  value 
is  not  known,  if  we  make  this  assumption,  I  say,  then  we 
may  distribute  these  380  losses  vertically,  throwing,  for 
instance,  18/64  of  them  into  the  seventh  row  and  so  on, 
and  so  also  with  the  other  numbers  24,  7  and  so  on,  and 
thus  produce  a  table  resembling  the  one  actually  obtained 
for  the  127  risks  upon  which  sound  value  was  given.  We 
should  then  know  for  each  risk  not  only  ratio  of  loss  to 
insurance  but  ratio  of  insurance  to  value  and  so  could 
obtain  the  desired  information,  namely,  the  ratio  of  loss 
to  value. 

This  procedure  of  course  would  be  almost  prohibitive 
because  of  the  labor  involved  and  as  a  matter  of  fact  it 
would  be  fruitless  to  follow  out  the  accidental  peculi- 
arities that  are  bound  to  occur  when  the  mass  of  statistics 
is  so  limited,  but  it  was  found  that  entirely  satisfactory 
results  could  be  obtained  if  instead  of  trying  to  match  the 
relative  distribution  in  each  separate  column  we  assumed 
the  relative  distribution  in  each  column  to  be  the  same  as 
the  general  distribution  got  by  summing  the  rows  hori- 
zontally, that  is  the  distribution  that  we  have  called  the 
law  of  insurance  to  value. 

This  is  not  a  rigorously  valid  assumption  for  if  risks 
on  which  the  ratio  of  loss  to  insurance  was  large  were  so 
distributed,  some  would  fall  well  up  in  the  table  where 
entries  fail  to  be  interpretable,  for  instance  there  could  be 
no  risk  on  which  at  the  same  time  the  ratio  of  loss  to  in- 
surance was  150  per  cent  and  the  ratio  of  insurance  to 
value  was  90  per  cent,  for  in  that  case  the  loss  would  be 


44 

135  per  cent  of  the  value,  which  is  manifestly  absurd ;  this 
was  remedied  in  a  somewhat  arbitrary  way  by  throwing 
down  and  back  into  the  table  the  few  entries  that  fell 
above  the  proper  limits. 

"While  this  whole  method  of  procedure  I  am  aware 
sounds  rather  arbitrary  when  thus  described,  as  a  matter 
of  fact  it  gave  a  distribution  which  was  an  enormous  im- 
provement over  the  crude  assumption  of  a  uniform  75  per 
cent  ratio  of  insurance  to  value,  which  in  itself  we  have 
recognized  as  not  so  altogether  bad  as  not  to  be  com- 
petent to  give  us  fairly  good  results. 

In  order  to  test  the  effect  of  this  method  of  distribu- 
tion, I  used  it  upon  the  127  risks  upon  which  sound  value 
was  given;  that  is  although  I  knew  the  actual  distribu- 
tion given  in  the  table  and  hence  the  desired  information, 
concerning  ratio  of  loss  to  value,  I  proceeded  from  the 
summation  numbers  64,  14,  8  and  so  on,  which  would  cor- 
respond to  the  sum  total  of  all  knowledge  obtainable  in 
the  cases  in  which  sound  value  was  not  given,  I  proceeded, 
I  say,  to  distribute  these  in  the  manner  described  and  thus 
to  reconstruct  the  table.  From  the  reconstructed  table 
ratios  of  loss  to  value  were  obtained  and  hence  a  table  of 
partial  loss ;  this  was  compared  with  the  table  of  partial 
loss  obtained  directly  from  the  statistics.  This  was  done 
for  two  classes  and  the  agreement  in  each  case  was  very 
close,  so  close  in  fact  as  to  serve  perfectly  well  as  a 
smoothed  expression  of  the  law  of  partial  loss. 

This  seemed  thoroughly  to  justify  the  method.  It  was 
therefore  used  upon  that  portion  of  the  statistics  upon 
which  sound  values  were  not  known  and  a  table  of  partial 
loss  for  these  risks  was  formed ;  this  was  then  united  with 
the  table  of  partial  loss  obtained  directly  from  the  statis- 
tics for  those  risks  upon  which  sound  value  was  given,  the 
two  of  course  properly  weighted,  and  thus  a  final  law  of 
partial  loss  was  obtained. 

As  a  matter  of  fact,  however,  there  were  other  diffi- 
culties to  be  dealt  with   before   these   results   could   be 


45 

actually  obtained.  Let  us  suppose  a  table  constructed  by 
the  method  described;  and  let  us  suppose,  for  instance, 
five  risks  upon  which  the  ratio  of  loss  to  insurance  is  be- 
tween 20  and  30  per  cent  and  the  ratio  of  insurance  to 
value  between  60  and  70  per  cent,  then  for  each  of  these 
five  risks  the  ratio  of  loss  to  value  is  somewhere  between 
12  and  21  per  cent ;  similarly  let  us  suppose  two  risks  for 
which  the  ratio  of  loss  to  insurance  is  between  70  and  80 
per  cent  and  the  ratio  of  insurance  to  value  between  80 
and  90  per  cent,  then  for  each  of  these  two  risks  the  ratio 
of  loss  to  value  will  be  between  56  and  72  per  cent.  Now 
similarly  we  should  obtain  a  great  array  of  intervals  for 
ratios  of  loss  to  value,  but  instead  of  intervals  bounded 
by  exact  numbers  of  tenths  we  should  have  a  great 
variety  of  overlapping  intervals  with  as  many  different 
boundaries  as  there  are  different  numbers  in  the  multi- 
plication table.    Such  confusion  would  be  hopeless. 

It  was  necessary,  therefore,  to  abandon  classification 
by  equal  intervals  and  to  substitute  a  set  of  intervals 
bounded  by  numbers  whose  products  all  belong  to  the 
same  system.  Such  a  system  of  numbers  is  formed  by  the 
powers  of  some  given  base.  In  this  case  the  base  8/10 
was  selected  and  the  intervals  from  (8/10)^"  to  1  were 
used;  the  risks  that  fell  below  (8/10)-*^  were  taken  all 
together.  This  had  also  the  advantage  over  the  decimal 
system  of  decreasing  the  size  of  the  intervals  toward  that 
end  of  the  table  at  which  the  data  were  more  numerous 
and  where  greater  refinement  of  treatment  was  neces- 
sary. 

This  method  of  treatment  proved  to  be  entirely  satis- 
factory except  that  it  involved  considerable  work,  par- 
ticularly in  the  final  passage  back  to  decimal  divisions 
which  was  necessary  in  order  to  obtain  a  usable  table  of 
partial  loss  and  also  in  the  carrying  out  of  certain  dis- 
tributions or  interpolations,  the  necessity  for  which  had 
perhaps  better  be  pointed  out.    A  certain  number  of  risks 


46 

are  found  to  have  a  ratio  of  loss  to  value  lying  between 
(8/10)^  and  (8/10)"  as,  for  instance,  the  risks  whose 
ratio  of  loss  to  insurance  lie  between  (8/10)^  and  (8/10)®, 
and  whose  ratio  of  insurance  to  value  lie  between  (8/10)^ 
and  (8/10)*;  a  certain  other  number  of  risks  are  found 
to  have  a  ratio  of  loss  to  value  lying  between  (8/10)"  and 
(8/10)",  as,  for  instance,  the  risks  whose  ratio  of  loss  to 
insurance  lie  between  (8/10)®  and  (8/10)^  and  whose 
ratio  of  insurance  to  value  lie  between  (8/10)^  and 
(8/10)*.  These  two  intervals  evidently  overlap  and  a 
distribution  is  necessary  among  the  three  intervals 
(8/10)«  to  (8/10)^  {S/lOy  to  (8/10)",  and  (8/10)"  to 
(8/10) ^\  The  method  of  distribution  actually  used  need 
not  be  discussed,  as  a  satisfactory  method  would  readily 
occur  to  anyone  who  handled  the  problem.  In  fact  the 
whole  statistical  treatment  is  too  full  of  detail  to  give  in 
full  in  this  report,  detail,  however,  which  would  readily 
be  supplied  by  anyone  who  undertook  to  work  with  the 
problem ;  I  have  attempted  to  give  only  an  outline  of  the 
method  employed. 

It  is  appropriate  at  this  point  while  the  tedious  com- 
plexity of  these  processes  is  freshly  in  mind  to  reflect  that 
they  are  made  necessary  only  by  the  fact  that  so  few  of 
the  proofs  of  loss  contain  sound  value.  The  device  that  I 
have  sketched  must  be  understood  to  be  an  expedient  that 
was  resorted  to  only  out  of  necessity.  The  question  then 
arises :  if  the  co-insurance  problem  is  worth  solving  is  it 
not  right  that  the  way  of  the  future  computer  should  be 
made  easier  and  the  result  more  perfectly  trustworthy  by 
insisting  that  proofs  of  loss  shall  always  contain  sound 
value?  Even  if  in  a  large  number  of  cases  the  values 
given  are  only  estimates,  the  estimates  will  be  far  better 
than  nothing,  for  when  dealt  with  in  a  mass  inaccuracies 
in  the  estimates  will  largely  neutralize  each  other. 

As  for  this  solution  I  may  say  that  there  has  been 
nothing  to  indicate  that  the  results  are  not  trustworthy, 


47 

in  fact  I  have  been  somewhat  surprised  to  find  how  in 
every  case  where  partial  checks  were  possible,  the  results 
stood  the  test. 

I  may  say  also  that  the  method  that  I  have  used  is  re- 
ducible to  mechanical  operations  that  can  easily  be  run 
through  by  an  ordinary  computer  and  the  difficulties  are 
not  greater  than  those  that  are  common  to  most  actuarial 
work. 

v.— The  Co-insurance  Problem  Treated  Algebraically 

Let  a  certain  interval  of  time  be  under  consideration,  pre- 
sumably one  year. 

Let  us  confine  our  attention  to  risks  that  all  belong  to  a 
single  class. 

Let  V  be  the  sound  value  of  each  risk. 

Let  N  be  the  number  of  risks  insured. 

Let  Lx  be  the  insurance  loss  on  these  N  risks  when  each 
risk  is  insured  for  Vio  of  its  value.  We  shall  later  consider 
the  case  in  which  the  risks  are  not  all  insured  for  the  same 
amount. 

Let  Ix  be  the  average  insurance  loss  per  risk.     1^  =      /m- 

Let  M  be  the  total  number  of  risks  upon  which  loss  is 
incurred. 

Let  m  X  be  the  number  of  risks  on  which  the  property  loss 
is  greater  than  Vio  V  but  less  than  ''+V10  V,  on  the  average 

a  /  ' 

(call  it)    /'  V;  then  mo  +  mi  -f- +  mg  =2  m,  =  M. 

Let  ti^  =™7m- 

Then  L,   =   mo  V  V  +  m^  Y  V+ . . .  . +^,.1  ^Vv 


Ao^   '^'^'^    /,o'-^""'^^^-'      / 


10 


+  VioV|m,  +  mx^i4-...m9|=M  V^iV,^•/-^Vloi)^l| 


48 


Let    s'/^iY-l-Vio  i/*i=Xx   ;  (2) 

0  /lO  X 

thenL^=MVX^,  (1) 

,      _  LJ     _    M/    ^ 


Let^/  =  J;  (4) 

then  Ix  =  J  V  K-  (3) 

Let  Rx  be  the  average  insurance  loss  per  dollar  of  insur- 
ance (hence  the  measure  of  the  hazard  or  net  rate)  when 
each  risk  is  insured  for  Vio  V. 

Let  Ix  be  the  total  amount  of  insurance  on  N  risks  when 
each  risk  is  insured  for  Vio  V; 

then  Ix  =  N  Vio  V  ; 

Let  y^^^  ^  /p,;  (6) 

then  Rx  =  J  Px>  (5) 

1^  =  JVX,  =  VioVPvx.  (7) 

Ix  is  the  expectation  of  insurance  loss  per  risk  ;  it  is  made 
up  by  (3)  of  two  factors,  J  and  VX^. 

J  is  the  probability  that  a  given  risk  will  become  a  claim  ; 
it  may  be  called  the  ignition  hazard.  VA-x  is  the  expectation 
of  insurance  loss  per  claim ;  it  is  made  up  by  (2)  of  ten  ele- 
mentary expectations.  The  product  of  VX^,  the  expectation 
per  claim,  by  J,  the  probability  that  a  given  risk  will  become 
a  claim,  is  the  expectation  per  risk  or  Ix- 

But  Ix,  instead  of  being  looked  at  as  the  sum  of  a  number 
of  elementary  expectations,  may  be  thrown  into  the  form  (7)  in 
which  it  is  an  equivalent  single  expectation  ;  the  amount  at 
stake  is  the  insurance,  VioV;  the  probability  of  its  being 
called  out  on  a  given  risk  is  Rx>  the  measure  of  the  hazard. 


49 

Rx  in  turn  is  made  up  of  two  factors,  J,  the  ignition  hazard, 
and  /3x  which  may  be  called  the  damage  hazard.  J  is  the 
probability  that  a  given  risk  will  become  a  claim ;  Px  is  the 
probability  that,  having  become  a  claim,  the  insurance  VioV 
will  be  called  out ;  p^^  is  the  measure  of  the  hazard  among  the 
claims,  R^  is  the  measure  of  the  hazard  among  the  risks. 

Neither  J  nor  p^  depend  upon  the  sound  value  V.  J  does 
not  depend  upon  the  ratio  of  insurance  to  value ;  /o^  however 
is  a  function  both  of  the  ratio  of  insurance  to  value,  since  it 
is  the  probability  of  insurance  loss,  and  of  the  damageability 
of  the  risk  as  given  by  the  /j-'s. 

This,  so  far,  is  all  under  the  assumption  that  the  insurance 
in  every  case  is  j  ast  VioV.  Let  us  now  suppose  that  the  N  risks 
are  not  all  insured  for  the  same  amount. 

Let  Ux  be  the  number  of  risks  insured  for  more  than  Vio  V 
and  less  than  ''+V10V,  on  the  average  (call  it)  yV  ;       then 

n^  +  Di  + +  ng  =2  nj  =  N. 

0 

Leti'x  =  °^'^. 

Just  as  Ix  is  the  insurance  loss  per  risk  when  there  is  an  in- 
surance of  Vio  V  so  let  I'x  be  the  insurance  loss  per  risk  when 

there  is  an  insurance  of    V  V.     In  general  let  primed  sym- 

/lO 

bols  refer  to  the  case  where  the   insurance  is    /    V.        The 

/lO 

expressions  for  this  case  are  the  same  as  those  already  given 
after  a  change  of  Vio  to    /     for  instance :    I'x  =     /    V  R'x, 

/lO,  /lO 

the  product  of  the  amount  at  stake,   namely   the  insurance, 
/    V,  and  the  measure  of  the   hazard,  R'^,  where  R'x=J/>'x 

/lO 

and  p'x  =    /u  As  a  matter  of  fact  X'x  may  with  sufficient 

'    Vio. 

accuracy  be  taken  to  be  Xx  +  X^  ^1 ,  although  a  closer  determi- 

2 

nation  might  be  made  if  it  were  thought  desirable. 


50 


Let  L  be  the  actual  insurance  loss  among  N  risks  insured. 
Let  us  use  actual  to  refer  to  the  case  where  the  distribution 
of  risks  as  to  amount  of  insurance  carried  is  described  by  the 
n's,  and  in  general  let  the  barred  symbols  refer  to  this  actual 
case. 

Then  L  =  n„l'o  +  nj/  + +  ng  I'g  =  2  nJi^}^J\ivp<.\. 

0  0 

9  _ 

Let  2  Vi  \\  =  X;  , 

then    L  =  N JVX  =  M Vx",  (8) 

1,  the  actual  average  insurance  loss  per  risk,  =  /{j=J  V  X. 

(10) 

T=noV V  +  n^y  V  + iigVv^N  V  I  i^.Y  . 

/lO  /lO  /lO  0         /lo 


9  \)/ 

2  V,   y    is  the  actual  average  ratio  of  insurance  to  value; 
call  this 


/lO 

b/. 


10 


Then  I  =  N  y  V. 

/lO 


^      L/        N  J  V  X 

/lO 

j/b/ 

Let  yb/-7; 

then   R  =  J  /3 

(12) 

(11) 
andT=  J  VX  =  y  VK:  (13) 

/lO 

1  is  in  reality  made  up  of  100  elementary  expectations  by 
(10),  (9)  and  (2)  but  it  reduces  by  (13)  to  an  equivalent 
single  expectation  in  which  the  amount  at  stake  is  the  actual 

average  insurance  /  V  and  the  probability   of  this  being 

called  out  is  K,  the  burning- ratio  or  ordinary  net  rate;  R 
again  is  the  product  of  the  ignition  hazard  J  and  the  damage 
hazard  p. 


51 


J  is  independent  of  the  v's  but  not  so  p. 

The  number  N  is  not  known  and  therefore  J  cannot  be 
found  directly.  The  damage  hazard  p  iiowever  can  be  com- 
puted and  if  the  ordinary  frate  R  has  already  been  ascer- 
tained J  may  be  computed  from  the  relation  K  =  J  /a. 

Since  R^  =  J  p^  and  K,  ==  J  /a,  Rx  =  R-'    /  This    gives 

the  coinsurance- rates  R^i^  terms  of  the  theoretically  correct 
single  rate  R.  Let  us  consider  however  instead  of  R  the 
rate  actually  in  use  R  which  may  or  may  not  be  correct.    Let 

then  R^  =  R  /—  These  rates  R^  will  produce  exactly  the 
same  income  as  the  single  rate  R,  for : 

the  income  is  2  Uj  /"   V  R' 

0         /lo 

.=  in,y  V  ^py 

0        /lo  /p 

NVll  ».     b,/    , 

p        0        /lo 
p  /lO 

If  R"  =  R,  this  income  is  the  expected  loss  L. 

This  equivalence  in  the  results  of  using  R^  and  R"  is  con- 
ditional however  upon  the  i^'s  remaining  the  same.      The  rates 

^^  rV 

Rx  will  produce  an  income  equal  to    /-^  times  the    expected 

/R 

loss  whatever  the  value  of  the  v's;  the  rate  IT  however  will 
produce  an  income  equal  to  =^times  the  expected  loss  where 

R*  is  determined  from  a  new  set  of  numbers,  v  *,  just  as  R  is 
determined  from  the  i^'s.  The  second  multiplier  is  a  function 
of  the  v*'s  while  the  first  is  not;  the  two  will  agree  in  general 
only  if  the  v**s  are  the  same  as  the  v's. 

tSince  R  is  independent  of  V  it  may  be  obtained  in  the  ordinary  way, 
without  any  assumption  as  to  V  by  dividing  the  entire  actual  insurance- 
loss  by  the  entire  actual  amount  of  insurance  in  force. 


52 

The  quantities  that  it  is  important  to  determine  are  the 
damage-hazards  /j^  and  p.  It  is  convenient  to  throw  the 
work  into  tabular  form  by  means  of  recursion  formulas,  and 
in  this  form  some  of  the  auxiliary  quantities  will  be  of 
interest. 

The  formulas  are  obtained  thus : 

I    X-l  9  1 

L'x  =  V/io  -j  2  a,  mi  -1-  X  2  m,  V  . 

Lx+i=  V/io  I  2  aj  mi  +  (x  +  1)  2mi  I  . 

Lx+i  =  Lx  -f  I    (ax— x)  m^  +  2  mi  Iv/io. 

Let  j  (a^— x)  m^-\-  2  mj  I  V/io  =  C^; 
then  L^^i  =  L,,  +  C^. 

9  9 

Let  2  mi  =  M^,  then  2  nij  =  Mx.i . 

X+l  X 

M^.i  =  Mx  +  m^. 
Furthermore  M9  =  o  and  Lo  =  0. 
By  these  formulas  the  values  L^  may  be  computed. 

'^x  =  i^  and  p,  =  y/_^, 

■v  ''     ^x    "T    ^x  +  1 

^  X 2~^' 

_^       9 

•\  =  2  I/,  X'l, 
0 

-    V- 

p=/h/- 

Ao 

For  actual  computation  we  may  conveniently  take  V  to  be 
one  hundred. 

The  formulas  then  are  : 
Mg  =  0. 
M^.i  =  Mx  +  m,. 


53 


a  =  l0|(a,-x)m, +  mJ 


Lo  =  0. 

Lx4-i  =  Lx  +  Cj 


100  m; 

^^"  /Vio- 

^  2 

9 

0 

Ao 

The  work  may  be  arranged  in  a  table  as  follows,  using 
for  an  example  the  figures  for  the  class  of  frame  mercantile 
buildings: 


54 


o 

• 

0 

TtH 

r-t 

iH 

• 

CD 

ci 

CO 

I-H 
I-H 

00 

T-H 
I-H 

CO 

q 

I-H 

CO 
q 

0:1 

10 

ifi 

0 

0 

0 

0 

0 

~J^ 

05 

01 

CO 

00 

0 

CD 

CO 

CO 

CO 

ci 

'^ 

0 

^ 

't 

CO 

CD 

t^ 

0 

03 

CO 

00 

"* 

Oi 

0 

I— I 

i-H 

I-H 

CO 
I-H 

CO 

lO 

0 

0 

0 

0 

10 

't 

IC 

0 

0 

CO 

I-H 

CD 

06 

t^ 

02 

cq 

■* 

1— 1 

10 

0 

CD 

t-^ 

0 

^ 

05 
1—1 

IM 

^ 

10 

I-H 

q 

t^ 

lO 

0 

0 

0 

0 

0 

'i^ 

10 

10 

iO 

iO 

f~) 

CD 

10 

CD 

l>^ 

^ 

0 

C5 

CD 

-t 

0 

i-O 

I-H 

t^ 

I-H 

^ 

CO 

(M 

cq 

0 

CD 

q 

CD 

10 

0 

0 

0 

0 

0 

10 

S 

10 

lO 

10 

0 

CO 

CO 

CO 

CD 

C2 

LO 

^ 

C5 

16 

CD 

iH 

l-H 
r-H 

CD 

CO 

CO 

rH 

Ci 

-* 

q 

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CO 

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Ci 

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0 

0 

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T^ 

'^^ 

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10 

IC 

CO 

CO 

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■* 

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r- 

^ 

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rH 

lb 

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0 

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q 

ro 

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M^ 

CO 

■* 

GO 

CO 

(M 

<M 

P 

0 

LO 

^ 

■* 

^ 

t^ 

t^ 

<N 

I-H 

I-H 

10 

• 

0<1 

10 

Tj5 

C5 

t-: 

CD 

t^ 

t^ 

TtH 

I-H 

CO 

lO 

CD 

r-l 

I~- 

10 

Q 

I— 1 

'f 

i-O 

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r-t 
CD 

05 

t^ 

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0 

0 

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CD 

10 

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■ 

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0 

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tH 

d 

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r— 1 

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0 

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^ 

0 

(M 

10 

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CD 

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(N 

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IC 

0 

0 

0 

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$s 

•^ 

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t^ 

r-- 

0 

CD 

1^ 

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<N 

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iC 

CD 

1—1 

t^ 

"P 

10 

ci 

cq 

0 

'^i 

IQ 

T-H 

<M 

0 

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« 

I-H 

05 

10 

C5 

I— ( 

(N 

(N 

<M 

Csl 

0 

0 

0 

0 

-* 

■^ 

Ci 

01 

<M 

'^i 

t^ 

0 

C<l 

rA 

CD 

r-l 

I-H 

ci 

P 

t^ 

CO 

CO 

t~- 

CO 

■* 

t^ 

(M 

Oi 

0 

IC 

I— 1 

r-l 

Ct 

CO 

I-H 

CO 

I-H 

05 
rH 
CO 

0 

CO 
I— 1 

CO 

CO 
CO 

0 
I— t 

>* 
1- 

I-H 

i-H 

CO 

0 
'*. 

CI 
Oi 

I-H 

CO 

0 

0 

H 

ksH 

<=; 

1 

>«; 

1 

a 

X 

1 

X 

3^ 

::^  J  ^ 

cS 

a 

X 

S 

O 


O 

o 

Oj 
O 
-«^ 
CQ 

•  rH 

a 

a 


o 


ri4 

O 

a> 
o 


a 

•  rH 

a 

a 

o 
o 


0} 

a 

o 


O 

d 


55 


Table  22.— The  Computatioit  for  the  Class  of  Frame 

Business  Buildings. 


X 

K 

^x  +  ^x+l 

^\ 

^x 

^x^'x 

0 
1 

.03200 
.07773 

.01600 
.03886 

0 
0 

0 

0 

.03200 

2 

.04573- 

.10097 

.05048 

2 

.000795 

3 

.05524 

.11739 

.05869 

5 

.002311 

4 

.06215 

.12948 

.06474 

1  4 

.007137 

5 

.06733 

.13868 

.06934 

1  4 

.007644 

6 

.07135 

.14589 

.07295 

2  9 

.016656 

7 

.07454 

.15169 

.07584 

21 

.012540 

8 

.07715 

.15645 

.07822 

2  2 
ITST 

.013550 

9 

.07930 

.16044 

.08022 

2  0 

.012633 

10 

.08114 

X    =       .073266 


/    the  actual  average  ratio  of  insurance  to  value,  is  found 

./  lOj 

to  be  .7077 

/lO 

By  interpolation  it  is  found  that  for  x  =7.3,  p^  =~p 
that  is  in  order  for  the  coinsurance  rates  to  produce  the  same 
income  as  the  ordinary  rate  the  73  per  cent  coinsurance  rate 
should  equal  the  ordinary  rate. 


•t-tAiujUs':.'?'" 


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W,UU  BE  A==^=,^°  ;°%"oUE  ?HB  PENALTY 
THIS  BOOK  °?'/"%°''J|nTS  ON  THE  FOURTH 
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